Cellularly-driven differences in network synchronization propensity are differentially modulated by firing frequency

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.     

On the Firing Rate Dependency of the Phase Response Curve of Rat Purkinje Neurons In Vitro

Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.     

Patterns in the discharge of simple and complex visual cortical cells

The activity of visual cortical neurons (area 17) was recorded in anaesthetized cats in response to sinusoidal drifting gratings. The statistical structure of the discharge of simple and complex cells has been studied as a function of the various parameters of a drifting grating: spatial frequency, orientation, drifting velocity and contrast. For simple cells it has been found that the interspike interval distributions in response to drifting gratings of various spatial frequencies differ only by a time scale factor. They can be reduced to a unique distribution by a linear time transformation. Variations in the spatial frequency of the grating induce variations in the mean firing rate of the cell but leave unchanged the statistical structure of the discharge. On the contrary, the statistical structure of the simple cell activity changes when the contrast or the velocity of the stimulus is varied. For complex cells it has been found that the invariance property described above for simple cells is not valid. Complex cells present in their activity in response to visual stimuli two different firing patterns: spikes organized in clusters and spikes that do not show this organization ('isolated spikes'). The clustered component is the only component of the complex cell discharge that is tuned for spatial frequency and orientation, while the isolated spike component is correlated with the contrast of the stimulus.     

Period doubling of calcium spike firing in a model of a Purkinje cell dendrite

Recordings from cerebellar Purkinje cell dendrites have revealed that in response to sustained current injection, the cell firing pattern can move from tonic firing of Ca²⁺ spikes to doublet firing and even to quadruplet firing or more complex firing. These firing patterns are not modified substantially if Na⁺ currents are blocked. We show that the experimental results can be viewed as a slow transition of the neuronal dynamics through a period-doubling bifurcation. To further support this conclusion and to understand the underlying mechanism that leads to doublet firing, we develop and study a simple, one-compartment model of Purkinje cell dendrite. The neuron can also exhibit quadruplet and chaotic firing patterns that are similar to the firing patterns that some of the Purkinje cells exhibit experimentally. The effects of parameters such as temperature, applied current, and potassium reversal potential in the model resemble their effects in experiments. The model dynamics involve three time scales. Ca²⁺- dependent K⁺ currents, with intermediate time scales, are responsible for the appearance of doublet firing, whereas a very slow hyperpolarizing current transfers the neuron from tonic to doublet firing. We use the fast-slow analysis to separate the effects of the three time scales. Fast-slow analysis of the neuronal dynamics, with the activation variable of the very slow, hyperpolarizing current considered as a parameter, reveals that the transitions occurs via a cascade of period-doubling bifurcations of the fast and intermediate subsystem as this slow variable increases. We carry out another analysis, with the Ca²⁺ concentration considered as a parameter, to investigate the conditions for the generation of doublet firing in systems with one effective variable with intermediate time scale, in which the rest state of the fast subsystem is terminated by a saddle-node bifurcation. We find that the scenario of period doubling in these systems can occur only if (1) the time scale of the intermediate variable (here, the decay rate of the calcium concentration) is slow enough in comparison with the interspike interval of the tonic firing at the transition but is not too slow and (2) there is a bistability of the fast subsystem of the spike-generating variables.     

Properties of human motoneurones and their synaptic noise deduced from motor unit recordings with the aid of computer modelling.

This paper reviews two new facets of the behaviour of human motoneurones; these were demonstrated by modelling combined with analysis of long periods of low-frequency tonic motor unit firing (sub-primary range). 1) A novel transformation of the interval histogram has shown that the effective part of the membrane's post-spike voltage trajectory is a segment of an exponential (rather than linear), with most spikes being triggered by synaptic noise before the mean potential reaches threshold. The curvature of the motoneurone's trajectory affects virtually all measures of its behaviour and response to stimulation. The 'trajectory' is measured from threshold, and so includes any changes in threshold during the interspike interval. 2) A novel rhythmic stimulus (amplitude-modulated pulsed vibration) has been used to show that the motoneurone produces appreciable phase-advance during sinusoidal excitation. At low frequencies, the advance increases with rising stimulus frequency but then, slightly below the motoneurones mean firing rate, it suddenly becomes smaller. The gain has a maximum for stimuli at the mean firing rate (the 'carrier'). Such behaviour is functionally important since it affects the motoneurone's response to any rhythmic input, whether generated peripherally by the receptors (as in tremor) or by the CNS (as with cortical oscillations). Low mean firing rates favour tremor, since the high gain and reduced phase advance at the 'carrier' reduce the stability of the stretch reflex.     

Alterations in the Intrinsic Burst Activity of Purkinje Neurons in Offspring Maternally Exposed to the CB1 Cannabinoid Agonist WIN 55212-2

Burst firing plays an important role in normal neuronal function and dysfunction. In Purkinje neurons, where the firing rate and discharge pattern encode the timing signals necessary for motor function, any alteration in firing properties, including burst activity, may affect the motor output. Therefore, we examined whether maternal exposure to the cannabinoid receptor agonist WIN 55212-2 (WIN) may affect the burst firing properties of cerebellar Purkinje cells in offspring. Whole-cell somatic patch-clamp recordings were made from cerebellar slices of adult male rats that were exposed to WIN prenatally. WIN exposure during pregnancy induced long-term alterations in the burst firing behavior of Purkinje neurons in rat offspring as evidenced by a significant increase in the mean number of spikes per burst (p < 0.05) and the prolongation of burst firing activity (p < 0.01). The postburst afterhyperpolarization potential (p < 0.001), the mean intraburst interspike intervals (p < 0.001) and the mean intraburst firing frequency (p < 0.001) were also significantly increased in the WIN-treated group. Prenatal exposure to WIN enhanced the firing irregularity as reflected by a significant decrease in the coefficient of variation of the intraburst interspike interval (p < 0.05). Furthermore, whole-cell voltage-clamp recordings revealed that prenatal WIN exposure significantly enhanced Ca²⁺ channel current amplitude in offspring Purkinje neurons compared to control cells. Overall, the data presented here strongly suggest that maternal exposure to cannabinoids can induce long-term changes in complex spike burst activity, which in turn may lead to alterations in neuronal output.     

Dynamics of the noisy neural network

The dynamic features of interspike interval sequences and structures of spatiotemporal patterns of firing in a coupled noisy neural network are investigated. The system displays complex dynamics under periodic external stimuli. The dynamics is modulated by a periodic impulse-like synaptic current which relates to a global coupling interaction between neurons. The firings of the stimulated neurons are phase-locked to this current. In addition, the interspike interval histograms are studied for the case of frozen noise which does not change its value within a time interval once it has been distributed onto the network. It is found that the peaks in these histograms are located at integer multiples of the period of the external stimulus, and the heights of these peaks decay exponentially, which corresponds to the experimental results. Received: 12 April 1997 / Accepted in revised form: 7 May 1997     

Effects of afterhyperpolarization on neuronal firing.

Stein's model for a neuron is studied. This model is modified to take into account the effects of afterhyperpolarization on the neuronal firing. The relative refractory phase, following the absolute one, is modelled by a time-increasing amplitude of postsynaptic potentials and it is also incorporated into the model. Besides the simulation of the model, some theoretical results and approximation methods are derived. Afterhyperpolarization tends to preserve the linearity of the frequency transfer characteristic and it has a limited effect on the moments of the interspike intervals in general. The main effects are seen at high firing rates and in the removal of short intervals in the interspike interval histogram.     

A Superposition Model of the Spontaneous Activity of Cerebellar Purkinje Cells

Based on physiological evidence for multiple firing zones in the dendritic arborizations of cerebellar Purkinje cells, a superposition model is proposed for spike triggering in these cells. Spike trains from 10 Purkinje cells were analyzed in terms of independence of interspike intervals and the properties of their variance-time curves. The results of this analysis were found consistent with the hypothesis that the spike train of a cerebellar Purkinje cell is the pooled output of a relatively large number of independent component processes. Simplifying assumptions as to the statistical nature of these processes lead to a very rough estimate of the number of firing zones.     

Predicting the Responses of Repetitively Firing Neurons to Current Noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.     

High-frequency organization and synchrony of activity in the purkinje cell layer of the cerebellum

The cerebellum controls complex, coordinated, and rapid movements, a function requiring precise timing abilities. However, the network mechanisms that underlie the temporal organization of activity in the cerebellum are largely unexplored, because in vivo recordings have usually targeted single units. Here, we use tetrode and multisite recordings to demonstrate that Purkinje cell activity is synchronized by a high-frequency (approximately 200 Hz) population oscillation. We combine pharmacological experiments and modeling to show how the recurrent inhibitory connections between Purkinje cells are sufficient to generate these oscillations. A key feature of these oscillations is a fixed population frequency that is independent of the firing rates of the individual cells. Convergence in the deep cerebellar nuclei of Purkinje cell activity, synchronized by these oscillations, likely organizes temporally the cerebellar output.     

Slow Oscillations of the Purkinje Cell Firing Rate Induced by Electrical Stimulation of the Locus Coeruleus in Rats

In anesthetized Wistar rats, we studied the effect of electrical stimulation of the locus coeruleus (LC) on the firing rates of Purkinje cells using spectral analysis. The frequency of extracellularly recorded activity of Purkinje cells was measured before and during the 1st, 5th, 6th, and 11th min after cessation of 10-sec-long LC stimulations. Spectral analysis of the Purkinje cell firing rates (imp./bin, the bin duration was 2^-8 sec) for 60- to 120-sec-long intervals was performed using fast Fourier transformation after digital conversion of unitary spikes. Mean power spectra of the Purkinje cell firing rates (derived from 8-sec-long consecutive epochs at a sampling rate of 256 sec^-1) showed an increase in the slow frequency range (0.1-1.0 Hz) after LC stimulation, particularly due to the slowest components (below 0.5 Hz). This effect lasted more than 1 min and usually less than 6 min after cessation of LC stimulation and could be interpreted as the development of slow oscillations in the Purkinje cell firing. Our results suggest that slow oscillations of the firing rate of cerebellar output neurons, induced by LC stimulation, reflect a specific coordination of the cerebellar neuronal activities (important for a central norepinephrine influence) in regulation of different pathological states.     

交流外电场下映射神经元放电节律的分析

神经元不同的放电节律承载着不同的刺激信息。文章基于神经元映射模型,研究低频交流电场对神经元放电节律的影响。在外部刺激下映射模型表现出丰富的放电模式,包括周期簇放电、周期峰放电、交替放电和混沌放电。神经元对刺激频率和振幅的变化极为敏感,随着频率的增大,放电节律表现出从簇放电到峰放电和混沌放电的反向加周期分岔序列;在周期节律转迁过程中存在一种新的交替节律,其放电序列为两种周期放电模式的交替,峰峰间期序列具有整数倍特征。外电场的频率影响细胞内、外离子振荡周期,导致神经元放电与刺激信号同步,对放电节律的影响更为明显。研究结果揭示了交流外电场对神经元放电节律的作用规律,有助于探寻外电场对生物神经系统兴奋性的影响和神经系统疾病的致病机理。     

Dynamical changes in neurons during seizures determine tonic to clonic shift

A tonic-clonic seizure transitions from high frequency asynchronous activity to low frequency coherent oscillations, yet the mechanism of transition remains unknown. We propose a shift in network synchrony due to changes in cellular response. Here we use phase-response curves (PRC) from Morris-Lecar (M-L) model neurons with synaptic depression and gradually decrease input current to cells within a network simulation. This method effectively decreases firing rates resulting in a shift to greater network synchrony illustrating a possible mechanism of the transition phenomenon. PRCs are measured from the M-L conductance based model cell with a range of input currents within the limit cycle. A large network of 3000 excitatory neurons is simulated with a network topology generated from second-order statistics which allows a range of population synchrony. The population synchrony of the oscillating cells is measured with the Kuramoto order parameter, which reveals a transition from tonic to clonic phase exhibited by our model network. The cellular response shift mechanism for the tonic-clonic seizure transition reproduces the population behavior closely when compared to EEG data.     

Variability measures of positive random variables

During the stationary part of neuronal spiking response, the stimulus can be encoded in the firing rate, but also in the statistical structure of the interspike intervals. We propose and discuss two information-based measures of statistical dispersion of the interspike interval distribution, the entropy-based dispersion and Fisher information-based dispersion. The measures are compared with the frequently used concept of standard deviation. It is shown, that standard deviation is not well suited to quantify some aspects of dispersion that are often expected intuitively, such as the degree of randomness. The proposed dispersion measures are not entirely independent, although each describes the interspike intervals from a different point of view. The new methods are applied to common models of neuronal firing and to both simulated and experimental data.     

No neural evidence for dynamic auditory tuning of the A1 receptor in the ear of the noctuid moth, <Emphasis Type="Italic">Noctua pronuba</Emphasis>

By examining the mechanical properties of the tympanum of the noctuid moth, Noctua pronuba, Windmill et al. (2006) suggested that this insect increases (up-tunes) the frequencies of its best hearing when exposed to high intensity sounds (HIS) resembling the echolocation calls of attacking bats. We tested whether this biophysical phenomenon was encoded in the neural responses of this moth’s most sensitive auditory receptor (A1 cell) before and after exposure to HIS. We measured: (1) the number of A1 action potentials (spikes) per stimulus pulse; (2) the proportion of A1 spike periods below that determined to elicit evasive flight maneuvers and, (3) the change in A1 cell firing (spike number, interspike interval, stimulus/spike latency) over a duration of time similar to that in which up-tuning lasts. We observed no significant spiking response changes in the predicted direction to any of the frequencies tested following exposure to HIS and we observed only two of the 24 predicted time-dependent changes to A1 firing. These results indicate that tympanal up-tuning does not result in a change to this moth’s auditory frequency sensitivity and we suggest either sensillar resonances or increases in thoracic muscle tension following exposure to HIS as alternative explanations.     

Phase resetting curves and oscillatory stability in interneurons of rat somatosensory cortex

Synchronous oscillations in neural activity are found over wide areas of the cortex. Specific populations of interneurons are believed to play a significant role in generating these synchronized oscillations through mutual synaptic and gap-junctional interactions. Little is known, though, about the mechanism of how oscillations are maintained stably by particular types of interneurons and by their local networks. To obtain more insight into this, we measured membrane-potential responses to small current-pulse perturbations during regular firing, to construct phase resetting curves (PRCs) for three types of interneurons: nonpyramidal regular-spiking (NPRS), low-threshold spiking (LTS), and fast-spiking (FS) cells. Within each cell type, both monophasic and biphasic PRCs were observed, but the proportions and sensitivities to perturbation amplitude were clearly correlated to cell type. We then analyzed the experimentally measured PRCs to predict oscillation stability, or firing reliability, of cells for a complex stochastic input, as occurs in vivo. To do this, we used a method from random dynamical system theory to estimate Lyapunov exponents of the simplified phase model on the circle. The results indicated that LTS and NPRS cells have greater oscillatory stability (are more reliably entrained) in small noisy inputs than FS cells, which is consistent with their distinct types of threshold dynamics.     

Detection and analysis of recurrent inhibition of firing motoneurons in man

During regular firing of "small" motor units, activated during weak voluntary contraction of the human soleus muscle, thick efferent fibers of n. tibialis were stimulated (a small M response was evoked, in which the small units did not participate). Peristimulus histograms of potentials of single motor units were constructed and the effect of stimulation on interspike interval duration was analyzed. The firing rate of the motor units was 4.5–7.6 spikes/sec. Stimulation of the nerve led to a sharp decrease in probability of their discharge or even complete temporary cessation of firing, i.e., it had a well marked inhibitory effect (lasting 10–20 msec). The latent period of inhibition (35–40 msec) was only a little longer than the latent period of the monosynaptic reflex of the soleus muscle. The effect of an inhibitory volley on duration of the interspike interval of the motor units depended on the time when the volley arrived during the interval. Lengthening of the interval was observed only if the inhibitory volley arrived in the second half or at the end of the interval. It is concluded that inhibition of firing of small motor units is due to Renshaw cells, activated on stimulation of axons of large motoneurons. The efficiency of a short (compared with the duration of the interspike interval) inhibitory volley reaching a motoneuron firing at low frequency characteristic of its adequate activation, is discussed.Institute for Problems of Information Transmission, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 16, No. 1, pp. 88–96, January–February, 1984.     

Partial phase synchronization of neural populations due to random Poisson inputs

We show that populations of identical uncoupled neurons exhibit partial phase synchronization when stimulated with independent, random unidirectional current spikes with interspike time intervals drawn from a Poisson distribution. We characterize this partial synchronization using the phase distribution of the population, and consider analytical approximations and numerical simulations of phase-reduced models and the corresponding conductance-based models of typical Type I (Hindmarsh-Rose) and Type II (Hodgkin-Huxley) neurons, showing quantitatively how the extent of the partial phase synchronization depends on the magnitude and mean interspike frequency of the stimulus. Furthermore, we present several simple examples that disprove the notion that phase synchrony must be strongly related to spike synchrony. Instead, the importance of partial phase synchrony is shown to lie in its influence on the response of the population to stimulation, which we illustrate using first spike time histograms.     

Bistability and the dynamics of periodically forced sensory neurons

Many neurons at the sensory periphery receive periodic input, and their activity exhibits entrainment to this input in the form of a preferred phase for firing. This article describes a modeling study of neurons which skip a random number of cycles of the stimulus between firings over a large range of input intensities. This behavior was investigated using analog and digital simulations of the motion of a particle in a double-well with noise and sinusoidal forcing. Well residence-time dis tributions were found to exhibit the main features of the interspike interval histograms (ISIH) measured on real sensory neurons. The conditions under which it is useful to view neurons as simple bistable systems subject to noise are examined by identifying the features of the data which are expected to arise for such systems. This approach is complementary to previous studies of such data based, e.g., on nonhomogeneous point processes. Apart from looking at models which form the backbone of excitable model s, our work allows us to speculate on the role that stochastic resonance, which can arise in this context, may play in the transmission of sensory information. Received: 22 March 1993/Accepted in revised form: 8 September 1993